We propose a method for numerical integration of Wasserstein gradient flows
based on the classical minimizing movement scheme.
In each time step, the discrete approximation is obtained as the solution
of a constrained quadratic minimization problem on a finite-dimensional function space.
Our method is applied to the nonlinear fourth-order Derrida-Lebowitz-Speer-Spohn equation,
which arises in quantum semiconductor theory.
We prove well-posedness of the scheme and derive a priori estimates on the discrete solution.
Furthermore, we present numerical results which indicate
second-order convergence and unconditional stability of our scheme.
Finally, we compare these results to those
obtained from different semi- and fully implicit finite difference discretizations.